Export unfinished image render
Yes, you can.  You will simply click the X beside the Render % or press Esc to cancel the render.  Then press F3 as usual and it will save the rendered portion to your desired format/location.  The unrendered part will be black if saved as a JPEG and transparent if saved as an Alpha PNG.

Just to prove it ;) here you go:



Here's the canceled render of the default scene saved as a JPEG:





EDIT: As was mentioned, Render Border can be useful in certain circumstances.  The problem is it's always rectangular (so it wouldn't have helped in this situation with a circular object) and it's usefulness is relegated to instances where you use multiple render layers.  Otherwise, you can simply crop the render dimensions as close to the edges of your scene as possible which does the same thing.